see for instance  [1]}, [2]} or [3]} for further details.
In particular, the derivative is uniquely defined, up to an additive constant.
In the rest of the paper, we will use the notation \(\partial _\xi h, \partial _\mu h\)  and \(\partial _\nu h\)  for the \(L\) -derivative of a function \(h\)  in the variable of the probability measure \(\xi \in \mathcal {P}_2(\mathbb {R}^{\ell }\times \mathbb {R}^m)\) , \(\mu \in \mathcal {P}_2(\mathbb {R}^{\ell })\)  and \(\nu \in \mathcal {P}_2(\mathbb {R}^{m})\) , respectively.
